der Hut vs die Mütze
Write "the hat" in German. Then they show a picture of a red baseball cap. So I write "die Mütze". Duolingo tells me I am wrong. I wouldn't mind so much if I could report that my answer should be accepted, but the only "report" option is "the picture does not match the given word". And there is no "discuss" button.
I am not German, so I am relying on dict.cc. They say:
hat [hat made of stiff material, particularly formal hats and/or hats with brims, e.g. sun hat, top hat, bowler hat] = Hut (m)
hat [hat or cap which is made of non-stiff material, e.g. bobble hat, cap] = Mütze (f)
Well, your description of the pictures matches another German word far better: We would call this "Kappe" or even use the English term "cap" for it..
However, this does not seem to be a solution for your duo answer.. Sorry. :-))
A Mütze is to keep your head warm. It is soft and, like gisberth wrote, you can put in your pocket without damage.
A Kappe is to have no sun in your face. It has a shield(?, German: Schirm) against the sun, but only in the front, not on the sides or in the back (but of cause you can turn it).
A Hut has a brim all around. For example Charlie Chaplin wears one. Today it is mostly seen as oldfashion to wear a Hut, or it is a sign of individual style.
Tipp, when you do not know what a noun exactly means: google it and let you show pictures!
My wife is German and laughed at me when I used "Hut" for hat explaining that it's more like a top hat/formal hat. Not sure why Duo/others always preset Hut as the main hat word.